an entire winning-set - he only needs to own a part of such a set. When can Breaker win in this case? Constant partial making m elements in one set (where Breaker does not own any element). If the size of each winning-set is at least m, and the number of sets is less than , then Breaker still has a winning strategy. The strategy uses a potential function: the potential of a "broken" set is 0, and the potential of a non-broken set E is , where r(E) is the number of elements Maker has to take in order to win it. So the initial potential of every winning-set is , and the potential of a set occupied by Maker is 1. From here the proof is the same as the Erdos-Selfridge theorem. Fractional making Suppose that, in order to win, Maker needs to own only a fraction t of the elements in one winning-set, where . So Breaker needs to own a fraction larger than (1-t) of the points in every set. Define the constant: (in the standard variant, ). If , then Breaker has a winning strategy when playing first. If , then Breaker has a winning strategy when playing second. In particular, if all sets are of size k and their number is less than , then Breaker (playing first) has a winning strategy. The strategy uses a potential function. The potential of a winning-set is defined as , where r is the number of elements Maker needs to take in order to occupy the set, and s is the number of elements Breaker needs to take in order to break it. If Maker occupies a set, then its potential will at some point be at least 1. Therefore, Breaker wins if he manages to keep the potential-sum below 1. Breaker's strategy is to take the element with the highest value, defined as the sum of potentials of winning-sets containing that element. Whenever Maker takes an element, the potential of every set containing it is multiplied by 2t, so it increases by (2t-1) times the current potential. Whenever Breaker takes an element, the potential of every set containing it is multiplied by (2-2t), so it increases by (1-2t) times the current potential. Whenever Breaker and Maker both touch the same set, its potential is multiplied by 2t(2-2t), so it increases by -(2t-1)2 times the current potential.